Tobler, W.

International Institute for Applied Systems Analysis (IIASA)

City Sizes, Morphology and InteractionTobler, W.IIASA Working PaperWP-75-0181975 Tobler, W. (1975) City Sizes, Morphology and Interaction. IIASA Working Paper. WP-75-018 Copyright © 1975 by the author(s). http://pure.iiasa.ac.at/410/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at CITY SIZES, MORPHOLOGY,AND INTERACTIONW. R. ToblerFebruary 1975 WP-75-l8Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor-mation purposes. The views expressedare those of the author, and do notnecessarily reflect those of IIASA.RM-77-2LINKING NATIONAL MODELS OF FOOD AND AGRICULTURE:An IntroductionM.A. KeyzerJanuary 1977Research Memoranda are interim reports on research being con-ducted by the International Institt;te for Applied Systems Analysis,and as such receive only limited scientifk review. Views or opin-ions contained herein do not necessarily represent those of theInstitute or of the National Member Organizations supporting theInstitute.On City Sizes and GrowthIt has frequently been observed thnt the built uparea of a city can be approximated very well Ｈ ｒ Ｒ ｾ .90) bya formula of the formwhere A is the built up area, 0< and S are numerical coefficients,and P is the number of people in the town. It follows that thepopulation density of a town isand that the rate of expansion of the town isClearly this relation can be inverted to estimate the pop-ulation from the growth in area. If a town is circular thenit follows that its radius isSr = ｾ p"2and its rate of radial growth isS - 1dr ｦ ｩ ｾ pi"dp = 2 'ITPrevious estimates of the coefficients have shownvariations in different cultures, Egyptian and Japanese townsbeing approximately sixteen times more compact than similarsized cities in West European cultures! (Tobler, 1969). Forthe latter Boyce (1963) and Nordbeck (1965) obtain approx-imatelyA = 0.003848 pO.88whereas Maher (1973) givesA = 0.001539 pO.87- 2 -for Canadian cities. In all of these casffithe estimates ofthe coefficients were obtained from cross section data. Itwas therefore considered worth examining at least one citythrough time. The attached qraph shows the logarithm ofthe area of Ann Arbor plotted against the logarithm of thepopulation. The area in this instance is the leqal area(in acres); this of course does not coincide with thefunctional built up area of the city. The growth has beencontinuous so that connecting the dots in population sizeorder also gives the temporal path, though not in equalincrements. For the last ｴ ｶ ｾ decades the data have beenavailable annually.The graph clearly splits into two periods,Pre 1'JNII:A = 1.0117 pO.2625,andPost V*HI:A = 0.0000029 pl.46252(A = km )where I have estimated the coefficients graphically. Bothperiods seem to display log-linear behavior, but ofdramatically different slopes. If this phenomenon isgenerally true perhaps this allometric relation can be usedto monitor structural changes in the urban system.VI!. Tobler,Laxenburg,4 February, 1975.oN..<::, ..........- 3 -LOGARITHM OF POPULATION0co..0)--..0::i'·0>..c:I; .W0:::(') c:I;0• .ClI rx-.0)o-s,-"'-0::r::E-<0 Hco 0:::•c:x:to 00...:l IoN·CJJ(..OJ· .f''''ｉ ｾ ｬC) I.(:.. -r--.---...---+--------+---,----+..-.---.. ＭｾＭ｟ｴＭ ..,.----+_--..,---Ｂ Ｇ ｴ ｾ Ｎ ｏ ｏ 8.80 9.GO 10. 1;.0 11 .. 2C 12J/) E.V)"/l\- 4 -ReferencesR. Boyce, "Changing Patterns of Land Use Consumption"The Professional G8ographer, 15, No.2, (1963)pp. 19-24.C .A. ｾ ｬ ｡ ｨ ･ ｲ Ｌ "Urban Porm and City 5i Z8: An Ontario Example",pp. 37-46, of L. Bourne, R. ｾ ｡ ｣ ｋ ｩ ｮ ｮ ｯ ｮ Ｌ andJ. Simmons, eo s., The Form of Cities in CentralCanada, Research Pub. #11, Department of Geography,University of Toronto 1973.S. Nordbeck, The Law of Allometric Growth, Discussion Paper#7, MICMOG, Ann Arbor, 1965.w. Tobler "Satellite Confirmation of Settlement SizeCoefficients", l'.rea Vol. 1, #3 (1969), pp. 30-34.The literature on this topic is summarized in Tobler(1969), Ｎ Ｒ ｅ ｾ ｣ ｩ ｴ ［ for persons not familiar with this work thetwo graphs below may be helpful. Also observe that one iso-lated person requires a radius of 35 meters, by the formula.US CITIESI " • .:. .-• _.: ". ,:,1 " ••••.•••Ｎ Ｚ ｾ Ｌ ［ Ｚ ｾ Ｃ ｾ Ｌ ｾ Ｇ ｩ Ｇ ...-to '",.. 'Source:'.;l -- 1jｃｾｹ PopUohooB. Berry, F. Horton, Geographic Perspectives onUrban Systems, Prentice Hall, 1970, p.6S.CANADIAN CITIES/' '.Source: C.A. Maher,1-- ＭＭＭＭＭＭＭＭＭＭＭｾＭＮＭＭＭＭＭＭＭＭＭＭＭ ---- ＮＭＭｾＭＭＭＭＭＭＭ .------.-r-- ---- . ｾ, j(, 1(\ C'l r:Telol urhop r·r:;('Itl!aflVrl (OOO's)(I)A Difficulty in Urban Morphology Description-2The density of population (persons kilometers ) withina city is describable as a decreasing function ofdistance from the center of the city; usually-brD(r) = Ae ,although Gaussian curves, Gamma functions, Besselfunctions, modified Pareto functions, cylindrical,conical, parabolic, and cosine functions, and severalothers, have also been proposed. There are someversions which treat radially asymmetric cases, andsome consider changes over time. A and b are numericalcoefficients.(II) The radius of a circular city is describable byR = et NSwhereN is the total population;R is the distance to the edge of the city fromits center;alpha and beta are numerical coefficients.(III) The equation in (I) does not define an edge for thecity. But (II) clearly specifies the dt$tance atwhich the edge occurs. Empirical evidence seemscompe,lling for both equations.(IV) If we know the rural density, call it F, then thiscan be used to locate the edge of the city, using I:- 2 -F 71 -bR= .t-'...eand thusR = -1 1n (!:)b Aand, alternatively, if we know n then the rural densitycould be calculated.For Paris, Bussiere givesN = 7,6000,000 peopleR = 27.5 kilometersA 54,892 km-2= personsb = 0.211Thus-2F = 165.77 persons kmFor cities Nordbeck givescI. = 0.0358 = 0.44Thus, from II,R = 37.29 kmand one would then have,. from I,-2F = 21.01 persons km •The two results do not seem:very compatible witheach other. If Bussiere's data are used then Nordbeck'scoefficient cI. can be re-ca1ibrated to obtain:Population Radius .fLParis 7,600,000 27.5 0,0258Lyons 850,000 14.3 0.0352Marseilles Ｘ Ｐ Ｐ ｾ Ｑ Ｐ Ｐ Ｐ 13.0 0.0329Toronto 1,700,000 20.0 0.0363Toulouse 340,000 5.0 0.0184Auxerre 28,000 1.8 0.0199- 3 -The agreement with Nordbeck's result is good in threeof the six cases. These also agree with the value of 3/8found by Stewart and Warntz.Bussiere also shows thatLimr -+ 00 orfor D(r) dr dewhich for his data on Paris yields a population of7,746,829 people, 2% greater than the actual value.This is the population outside his "edge" for theregion of Paris. In his other cases he over or underestimates the total populations by comparable amounts,except for Toronto, where the population beyond theedge amounts to ten percent of the total. He doesnot make clear how he chose the edges for the urbanregions which he studied. This is a pity becausehis is otherwise one of the outstanding treatmentson the topic. Presumably Bussiere takes the "edge"of the city to be the point at which the cumulativepopulation curve begins to fluctuate erratically.More attention needs to be paid to the boundary.Stewart and Warntz also assert that the centraldensity of a city, A, is equal to12"A = k • N •Using Bussiere's data the constant k becomes forParis k = 19.9Lyons 29.9Marseilles 47.3Toronto 8.0Toulouse 53.9Auxerre 54.8A rather variable constant. Winsborough claims thatA is a function of the age and type of city.It has also been asserted (e.g. Weiss) that theexponent b is a function of the size of the city (C = 1/2):-cb = aN •A logarithmic plot of Bussieres'data shows thisrelationship. If A, b, and R were all such simple functionsof N, then this one number would completely specify themorphology of a city. But they appear to vary according tothe geographical context, being relatively stationary withinlarge culture regions, but still depending on the geographicallocation relative to all other place in the same region.But Bussiere has recently shown that A and b Plot as astraight line over time for several cities. Since they arerelated in the following way:N = 2nAb 2these graphs are a type of phase diagram for urban growth.Thus some improvement is being made in our empiricalunder standing of urban morphology.W. Tobler,Laxenburg,20th February, 1975.••ｾ ｾ ,\)ｾ • -.t•Q-!ｾ ｾ., f't-..-'"• ｾｾ ｾd.- E'ｾ""--'"ellｾ...0........•......o----..,..--------..,..---------+!N8REF ERE N C E SRene Bussiere, The Spatial Distribution of Urban Population,Paris, Centre de Reserche d'Urbanisme, 1970.R. Bussiere, "Static and Dynamic Characteristics of theNegative Exponential Model of Urban PopulationDistributions", Patterns and Processes in Urban andRegional Systems, A. Wilson, ed., London, Pion Press,1972.Stig. Nordbeck, The Law of Allometric Growth, Discussion PaperNo.7, Ann Arbor, MICMOG, 1965.J. Stewart, W. Warntz, "The Physics of Population Distribution",Journal of Regional Science, 1 (1958).H. Weiss, "The Distribution of Urban Population," OperationsResearch, 9 (1961) pp.860-874.H. Winsborough, "City Growth and City Structure", Journal ofRegional Science, 4 (1963).A Re-examination of Soroe Old Data on Linguistic Borders andSpatial InteractionThe classic study of the effects of a linguisticborder on the amount of interaction between geographic areaswas published some years ago by MacKay (I). More recent workon this topic has been presented inter alia, by Gould andWhite (2). The present note is based on data assembled, andalso analyzed, by Lill (3). The results are not as definativeas I had hoped, but the data were too enticing, and I couldnot resist a quick look. Tables I and II give the number ofpassengers travelling on a portion of the Austrian Nordwestbahnduring the year 1889. The selected section of the line runsNNW from Vienna past Prague to the then border of the Austro-Hungarian empire near Tetschen; see Figure I. The underlinedplaces were asserted by Lill to be Czech speaking. The locationsnot underlined were German speaking. ｾ ｩ ･ ｮ ｮ ｡ was, ofcourse, cosmopolitan and a goodly proportion of itsinhabitants spoke Ceech. LillIs analysis of thesedata, he concludes, demonstrates a propensity to interactwhich is greater between linguistically similar places. Buthe does not do this in quite the same way as do MacKay orGould and White. The thrust of LillIs paper, of course, isa thorough derivation, verification and application of whatwe now call the gravity model of interaction. Only a smallsection of the work is devoted to the study of linguisticeffects. What I have done here is to have replotted some ofhis data in the currently fashionable manner. All of thedata in the acoompanying tables are extracted from LillIspaper. The greatest shortcoming of the data, I believe, isthat they represent only the passengers of the one railroadcompany. For example, there existed another railroadconnecting Prague and Vienna, to the west of the line shownin Figure I. LillIs original map shows several of the otherpossible routes of travel. Unfortunately the passengercounts for these other lines are not available. From Viennato Kolin the line shown seems to have been the most directconnection. An asterisk in table I indicates places which arejunction points, as far as I can tell from LillIs map.- 2 -The present border between Austria and ｃ ｺ ･ ｣ ｨ ｯ ｳ ｬ ｯ ｶ ｾ ｫ ｩ ｡bisects the section Retz-Znaim; my touristic map shows nocurrent railroad connecting the two. Since these borderswere supposed to have been based on local preferences, itis to be asked whether LillIs lincruistic knowledge wascorrect. The borders were also supposed to end wars:clearly they were a failure! I have not examined any of thedata collected by the Hilson peace commissions, though itshould be available in some archive. They may even haveused interaction data as a part of their deliberations.A plot (Figure II) of a portion of the data fromTable I suggests that Gross Priesen and Tetschen really wereGerman speaking. The anomolous values for Gross Nossek,Nimburg, and perhaps some others, may be explainable by thestatus of these as transfer stations, connecting to trainsgoing to other parts of the empire. I have not sketchedregression lines, so as to avoid biasing the reader's eye.Interestingly the slopes of the two lines which I woulddraw are different, and are not just a spatial translation.Znaim would fall at the intersection of my lines, suggestingthat it is indeed a border town. The rate of decay ofpassenger traffic appears steeper beyond this point,especially if Gross Priesen and Tetschen are dis-regarded.Analysis of covariance, discriminant analysis, and so on,are statistical techniques of greater elaboration which maybe used on these data. I find it difficult to detect anyeffect other than distance decay in the figure givinginteraction with Prague.ｾ Ｑ Ｎ Tobler,Laxenburg,30 Jan. 19 75 .TABLE I1890Popula-tionFrom RR Passenqersｾ ｖ ｩ ｣ ｮ 1889 ..IRR km From Wien To WienHien *Jed1esee *KorneuburgStockerauOberho11abrunnRetzZnaim (Znojmo) *vｓ｣ｨｾｮｷ｡Ｑ､Ｍｆｲ｡ｩｭ (Sumna)M. Budwitz (Moravske Eudejovice)Wiese (Luka n Jih1avou)Ig1au (Jih1ava) *Po1na.'" .Deutschbrod (Hav11ckuv Brod) *Swet1a (Svet1a n Sazavou)Go1tsch Jenikau (Go1cou Jenikov)v.,..Cas1au (Cas1av)Sed1etz Kuttenberg (Kutna Hora)Kolin *Gross Nossek *Podebrod (Podebrady)Nimburg (Nymburk) *Lissa (Lysa n Labem) *Wschetat Privor (Vsetaty) *He1nikPo1eppLeitmeritz Stadt Ｈ ｌ ｩ ｴ ｯ ｭ ･ ｲ ｩ ｾ ･ ＩAussig (Usti n Labem) *Gross PriesenTetschen (Decin)705,4002,31214,24514,24014,08111,20431,80320,01028,75010,42451,04918,71629,66621,48916,03533,24743,94435,1871,89712,70111,9379,8016,08422,8518,77125,43810,5836,00027,825o61523528510312113918619821022524126727928929930931532333836337540040943444445837,798103,75866,39921,9418,94319,3032,0554,6656237,7268382,5854273506815555302151285281491399315351,040208,846105,25766,72522,55018,2405,5308,9972,6339286556183465026961,2206,618--PR Passencrers 1889PP kInTl>..BLE I I From PopulationPrao FroJT1 Prag To Prag 1890Praa 0 -- -- 205,100-----'-Kolin 74 822 1,301 35,187s. Kuttenberg 85 5,856 6,773 43,944Cas1au 95 6,234 6,273 33,247G. Jenikau 106 1,852 -- 16,035swet1a 132 1,087 -- 21,489Deutschbrod 149 1,579 j 1,762 29,666Po1na 163 342 -- 18,716Ig1au 175 1,506 1,585 51,049\'liese 187 23 -- 10,424M. Budwitz 234 149 139 28,750ｓ ｣ ｨ ｾ ｮ ｷ ｡ Ｑ ､ Ｍ ｆ 252 11 -- 20,010Znaim 274 265 309 31,803Retz 288 14 -- 11,204Oberho11abrunn 321 14 15 14,081Stockerau 350 21 22 14,240Korneuburg 358 27 46 14,245Jed1esee 367 4 -- 2,312Wien 373 1,596 1,647 705,400A Ｎｳ･ＮｃＮｾｉﾷｏｴ｜ 0 { {he.Ｐ ･ Ｕ Ｋ ｣ ｾ ｲ ･ ｜ ｣ ｨ ｾ ｾ ｣ Ｎ ｨ ･ Ncr4w e"',* bA.hn) ＱＡｬｾｱａｾｾ･Ｂ L-ll tI SO ｫｾ:5. ｋｾ ｴｴ･ｬＧ｜｢｡ｾ｡ＮﾭCel.::.lal.A;;ｓｷ･ｾｬ｡ＮＮDe14#-scl. b...."'c::lｓｴｯｫ･ｩＧｾ｜ａＬ. k.orl'ltu.bu ..ｾ. • \ 9 Jed lesu'NI E.'" 0<lｾ, ｾeu:> )( xｾ x\d '1 <'I4S.4ｾＬﾧ 4ｾ.J ｾ ｾ)< xLC 13""ｾ ,(xX•o••o.....,"T"....----------.. \ ｾＮ｟ＭｾＭＭＬｾＭＮＭＢ ...... ｾＭＭＬＭＭＭＭＭＭＮＭＮＭＭＮ --1""" ｾ {'W<? J S Y "'PI ｾ c/ ':J J f:/ / 0 ｉｾ ＧＱＱＯｾ tJ to7J:s •\I? •d- e •=£ •. -.3[=' ( L.0 0\) 1ｾｾ <S'">l\if..i: ｾd5<r7e0:.fQｾVJｾｾ-ｾE0- .....Y..ｾ!).... jｾＮｾｾ-.rU"l<J- 3 -Acknowledgements:My thanks to E. ｕ ｾ ｳ ･ ｲ for procuring the Lill paper and toJ-M Gambrelle for converting and plotting the data.Re fe rences:J. R. Mackay, "'l'he In te ractance Hupothesis and Boundaries inCanada: A Preliminary Study" Canadian Geographer 11 (1958)pp. 1-8.P.R. Gould, and R. White, Mental Maps, London, Penguin Books1974.Edward Lill, Das Reisegesetz und seine Anwendung auf denEisenbahnverkehr mit verschiedenen auf die Betriebsergebnissedes Jahres 1889 bezugnehmenden statistische BeilaqenTabellen und Bildlicher Form, Wien, Spielhagen und Schurich,1891, 44 pp. + tables.

City Sizes, Morphology and Interaction

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